Computer Analysis of Sprouts

Abstract

Sprouts is a two-player pencil-and-paper game with a
topological flavor. It was invented in 1967 by Michael Paterson and
John Conway, and was popularized by Martin Gardner in the Mathematical
Games column of Scientific American magazine.

We have written a computer program to analyze the n-spot game of
Sprouts for general n. Our program uses a number of standard
techniques to expedite adversary searches such as cutting off the
search as soon as the value can be determined, and hashing previously
evaluated positions. But the truly innovative feature is our
representation of game positions, which provides enough information to
generate moves and has the property that many different planar graphs
collapse into the same representation. This has an enormous impact on
the speed of the search.

The complexity of n-spot Sprouts grows extremely rapidly with n.
According to Gardner, Conway estimated that analysis of the
eight-spot game was beyond the reach of present-day computers. Before our
program, even the value of the seven-spot game was unknown; we have
calculated the value of all games up to and including eleven spots. Our
calculation supports the Sprouts Conjecture: The first
player loses if n is 0, 1 or 2 modulo 6 and wins otherwise.